mirror of
https://github.com/allaunthefox/Research-Stack.git
synced 2026-07-31 03:05:21 +00:00
540 lines
24 KiB
Markdown
540 lines
24 KiB
Markdown
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Authors: Research Stack Team
|
||
|
||
SSMS.lean — Scalar-Spawning Manifold State Machine
|
||
|
||
Full Lean 4 formalization covering:
|
||
§1 Q16.16 fixed-point arithmetic
|
||
§2 Ternary weights and dot product
|
||
§3 BitLinear activation scaling
|
||
§4 MLGRU recurrent state
|
||
§5 Scalar node state machine
|
||
§6 SUBLEQ core and step semantics
|
||
§7 N-gossip protocol
|
||
§7.5 Phantom coupling (J_phantom cost)
|
||
§8 Directed simplicial complex
|
||
§9 Betti Swoosh Hamiltonian H_M(t) = −Δ_M + V_M
|
||
§10 Anti-Collision Identity (ACI) and preservation theorem
|
||
§11 SRAM banking layout and conflict-free theorem
|
||
|
||
Per AGENTS.md §1.4: All new hot-path code uses Q16_16 fixed-point.
|
||
Per AGENTS.md §2: All code uses PascalCase for types, camelCase for functions.
|
||
-/
|
||
|
||
import Std
|
||
import Mathlib.Tactic.NormNum
|
||
import Semantics.Timing
|
||
|
||
import Semantics.FixedPoint
|
||
import Semantics.Tactics
|
||
|
||
namespace Semantics.SSMS
|
||
|
||
open Semantics
|
||
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §2 Ternary Weights
|
||
-- w̃ ∈ {−1, 0, +1} stored as 2-bit codes, 16 per 32-bit word.
|
||
-- Dot product: only ADD/SUB, no MUL co-processor.
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
inductive TernaryWeight where
|
||
| Pos : TernaryWeight -- code 01 → +1
|
||
| Zero : TernaryWeight -- code 00 → 0
|
||
| Neg : TernaryWeight -- code 10 → −1
|
||
deriving Repr, DecidableEq, Inhabited
|
||
|
||
def TernaryWeight.toQ : TernaryWeight → Q16_16
|
||
| .Pos => Q16_16.one
|
||
| .Zero => Q16_16.zero
|
||
| .Neg => Q16_16.negOne
|
||
|
||
/-- Number of 32-bit words needed to store d ternary weights (2 bits each). -/
|
||
def wordsNeeded (d : Nat) : Nat := (d + 15) / 16
|
||
|
||
/-- Ternary weight slice for one scalar: d weights as two Boolean arrays.
|
||
Disjoint invariant: no weight can be simultaneously +1 and −1. -/
|
||
structure TernarySlice (d : Nat) where
|
||
wPos : Array Bool -- wPos[j] = true ↔ w̃ⱼ = +1
|
||
wNeg : Array Bool -- wNeg[j] = true ↔ w̃ⱼ = −1
|
||
sizePos : wPos.size = d
|
||
sizeNeg : wNeg.size = d
|
||
disjoint : ∀ j : Fin d,
|
||
¬ (wPos[j]'(sizePos ▸ j.isLt) ∧ wNeg[j]'(sizeNeg ▸ j.isLt))
|
||
|
||
/-- Ternary dot product: Σⱼ w̃ⱼ · xⱼ.
|
||
Weight=+1 → ADD xⱼ (2 SUBLEQ).
|
||
Weight=−1 → SUB xⱼ (1 SUBLEQ).
|
||
Weight= 0 → NOP.
|
||
No MUL co-processor calls. -/
|
||
def TernarySlice.dot {d : Nat} (ws : TernarySlice d) (xs : Fin d → Q16_16) : Q16_16 :=
|
||
(List.range d).foldl (fun acc j =>
|
||
if hj : j < d then
|
||
let _p := ws.wPos.getD j false
|
||
let n := ws.wNeg.getD j false
|
||
let x := xs ⟨j, hj⟩
|
||
Q16_16.add acc (if _p then x else if n then Q16_16.neg x else Q16_16.zero)
|
||
else acc
|
||
) Q16_16.zero
|
||
|
||
/-- Memory compression ratio: 2 bits/weight vs 32-bit Q16.16. -/
|
||
theorem compressionRatio : (2 : Rat) / 32 = 1 / 16 := by norm_num
|
||
|
||
/-- Against FP16 baseline (16-bit): 2 bits/weight → 8× reduction.
|
||
With activation savings: total ≈ 0.1× M_FP16. -/
|
||
theorem fp16Compression : (2 : Rat) / 16 = 1 / 8 := by norm_num
|
||
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §3 BitLinear Activation Scaling
|
||
-- x̃ = Clip(x · α, −Q_b + ε, Q_b − ε)
|
||
-- α = Q_b / (η + ε), η = max{|xᵢ|} (butterfly MAX)
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
structure BitLinearParams where
|
||
qB : Q16_16 -- quantization range: 128 for 8-bit = 0x00800000
|
||
eta : Q16_16 -- global abs-max from butterfly MAX reduction
|
||
alpha : Q16_16 -- = qB / (eta + ε), computed via NR reciprocal
|
||
|
||
def BitLinearParams.compute (qB eta : Q16_16) : BitLinearParams :=
|
||
{ qB
|
||
eta
|
||
alpha := Q16_16.mul qB (Q16_16.recip (Q16_16.add eta Q16_16.epsilon)) }
|
||
|
||
/-- Scale activation and clip to quantization range. -/
|
||
def bitLinearScale (p : BitLinearParams) (x : Q16_16) : Q16_16 :=
|
||
Q16_16.clip
|
||
(Q16_16.mul x p.alpha)
|
||
(Q16_16.add (Q16_16.neg p.qB) Q16_16.epsilon)
|
||
(Q16_16.sub Q16_16.epsilon p.qB)
|
||
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §4 MLGRU Recurrent State
|
||
-- hₜ = fₜ ⊙ hₜ₋₁ + (1 − fₜ) ⊙ cₜ
|
||
-- MatMul-free: fₜ and cₜ from ternary dot products.
|
||
-- Only 2 MUL co-processor calls for the gating blends.
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
structure MlgruState where
|
||
hT : Q16_16 -- current hidden state
|
||
hPrev : Q16_16 -- previous (for Δh gossip trigger)
|
||
deriving Repr, Inhabited
|
||
|
||
/-- One MLGRU recurrence step.
|
||
fT: forget gate (from ternary dot product, Q16.16).
|
||
cT: candidate state (from ternary dot product, Q16.16). -/
|
||
def mlgruStep (fT cT : Q16_16) (st : MlgruState) : MlgruState :=
|
||
let termA := Q16_16.mul fT st.hT -- fT · h_{t-1}
|
||
let oneMf := Q16_16.sub fT Q16_16.one -- 1 − fT
|
||
let termB := Q16_16.mul oneMf cT -- (1 − fT) · cT
|
||
{ hT := Q16_16.add termA termB, hPrev := st.hT }
|
||
|
||
/-- Hidden-state update magnitude — primary spawn signal in recurrent mode. -/
|
||
def MlgruState.delta (st : MlgruState) : Q16_16 :=
|
||
Q16_16.abs (Q16_16.sub st.hPrev st.hT)
|
||
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §5 Scalar Node State Machine
|
||
-- Sᵢ = (sᵢ, σᵢ, eᵢ, hidden, ver, load)
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
structure ScalarNode where
|
||
s : Q16_16 -- scalar value (= hT after MLGRU closes the loop)
|
||
sigma : Bool -- activation status: true = active, false = dormant
|
||
energy : Q16_16 -- gradient energy eᵢ = ‖∂L/∂sᵢ‖₂ Q16.16
|
||
hidden : MlgruState -- MLGRU recurrent state
|
||
version : Nat -- gossip version counter
|
||
load : Q16_16 -- work-queue depth |Wᵢ|
|
||
deriving Repr, Inhabited
|
||
|
||
/-- Spawn condition: eᵢ ≥ τ_spawn. -/
|
||
def ScalarNode.shouldSpawn (nd : ScalarNode) (τ : Q16_16) : Bool :=
|
||
decide (τ ≤ nd.energy)
|
||
|
||
/-- Fold condition: eᵢ ≤ τ_fold. -/
|
||
def ScalarNode.shouldFold (nd : ScalarNode) (τ : Q16_16) : Bool :=
|
||
decide (nd.energy ≤ τ)
|
||
|
||
/-- Transition with hysteresis (prevents oscillation at threshold).
|
||
Spawn wins over fold when both conditions hold. -/
|
||
def ScalarNode.transition (nd : ScalarNode) (τSpawn τFold : Q16_16) : Bool :=
|
||
if nd.shouldSpawn τSpawn then true
|
||
else if nd.shouldFold τFold then false
|
||
else nd.sigma
|
||
|
||
/-- Current rank: number of active scalars in pool. -/
|
||
def poolRank (nodes : Array ScalarNode) : Nat :=
|
||
nodes.foldl (fun acc nd => if nd.sigma then acc + 1 else acc) 0
|
||
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §6 SUBLEQ Core and Step Semantics
|
||
-- Single instruction: M[b] ← M[b] − M[a]; if M[b] ≤ 0: PC ← c
|
||
-- Negative addresses are memory-mapped ports (ports §2 in §1).
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
/-- One SUBLEQ instruction. -/
|
||
structure Subleq where
|
||
a : Int -- source address (negative = mapped port)
|
||
b : Int -- destination address
|
||
c : Int -- branch target when M[b] ≤ 0 after subtract
|
||
deriving Repr
|
||
|
||
abbrev Program := Array Subleq
|
||
|
||
structure SubleqCore where
|
||
mem : Int → Q16_16 -- full address space; M[-1..M[-22] = ports
|
||
pc : Nat
|
||
program : Program
|
||
|
||
/-- Single deterministic step. -/
|
||
def SubleqCore.step (core : SubleqCore) : SubleqCore :=
|
||
if h : core.pc < core.program.size then
|
||
let ⟨a, b, c⟩ := core.program[core.pc]'h
|
||
let result := Q16_16.sub (core.mem a) (core.mem b) -- matched subleqOp
|
||
let mem' := fun addr => if addr == b then result else core.mem addr
|
||
let pc' := if result.toInt ≤ 0 then c.toNat else core.pc + 1
|
||
{ core with mem := mem', pc := pc' }
|
||
else core
|
||
|
||
/-- Run for exactly n steps (deterministic, no fuel ambiguity). -/
|
||
def SubleqCore.runN (core : SubleqCore) (steps : Nat) : SubleqCore :=
|
||
Nat.rec core (fun _ acc => SubleqCore.step acc) steps
|
||
|
||
/-- Halt predicate: PC beyond program length. -/
|
||
def SubleqCore.halted (core : SubleqCore) : Bool :=
|
||
decide (core.pc ≥ core.program.size)
|
||
|
||
-- Memory-mapped port addresses (standard across all scalar nodes).
|
||
namespace Ports
|
||
def ioIn : Int := -1
|
||
def ioOut : Int := -2
|
||
def sVal : Int := -3 -- scalar value sᵢ
|
||
def sigmaPort : Int := -4 -- activation flag
|
||
def energyPort : Int := -5 -- gradient energy eᵢ
|
||
def tauSpawn : Int := -6
|
||
def tauFold : Int := -7
|
||
def mulA : Int := -8 -- co-processor factor a
|
||
def mulB : Int := -9 -- co-processor factor b
|
||
def mulResult : Int := -10 -- co-processor result (1-cycle latency)
|
||
def gossipOut : Int := -11
|
||
def gossipIn : Int := -12
|
||
def hTPort : Int := -13 -- MLGRU hidden state
|
||
def fGate : Int := -14
|
||
def cTPort : Int := -15
|
||
def etaPort : Int := -16 -- abs-max from butterfly MAX
|
||
def alphaPort : Int := -17 -- qB / (η + ε)
|
||
def wPtr : Int := -18 -- ternary weight base address
|
||
def wPosPort : Int := -19 -- current +1 bitmask word
|
||
def wNegPort : Int := -20 -- current -1 bitmask word
|
||
def etaOut : Int := -21 -- emit |sᵢ| for butterfly MAX
|
||
def etaIn : Int := -22 -- receive global η
|
||
def frustPrevX : Int := -23 -- stores P_{m-1} coordinate
|
||
def frustAniso : Int := -24 -- Anisotropy Tensor A_ij
|
||
def frustResult : Int := -25 -- returns I_lock(X - prevX, A)
|
||
end Ports
|
||
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §7 Modified N-Gossip Protocol
|
||
-- Fanout: n_contact = ⌈log₂ K⌉
|
||
-- Stratified: ⅓ hot (high Δh), ⅓ cold (low Δh), ⅓ random
|
||
-- Update: eᵢ ← max(eᵢ, eⱼ) (spawn-biased)
|
||
-- Anti-entropy: version-vector repair of lost gradient fragments
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
/-- Full gossip packet (all numerics Q16.16). -/
|
||
structure GossipPacket where
|
||
energy : Q16_16
|
||
sigma : Bool
|
||
sVal : Q16_16
|
||
version : Nat
|
||
load : Q16_16
|
||
deltaH : Q16_16 -- |hₜ − hₜ₋₁|: recurrent spawn signal
|
||
deriving Repr, Inhabited
|
||
|
||
def ScalarNode.toGossip (nd : ScalarNode) : GossipPacket :=
|
||
{ energy := nd.energy
|
||
sigma := nd.sigma
|
||
sVal := nd.s
|
||
version := nd.version
|
||
load := nd.load
|
||
deltaH := nd.hidden.delta }
|
||
|
||
/-- Merge: propagate maximum energy, increment version. -/
|
||
def ScalarNode.gossipMerge (nd : ScalarNode) (pkt : GossipPacket) : ScalarNode :=
|
||
let e' := if pkt.energy > nd.energy then pkt.energy else nd.energy
|
||
let _δh' := if pkt.deltaH > nd.hidden.delta
|
||
then pkt.deltaH else nd.hidden.delta
|
||
{ nd with energy := e', version := nd.version + 1 }
|
||
|
||
/-- Fanout: contacts per gossip round = ⌈log₂ K⌉. -/
|
||
def nContact (K : Nat) : Nat :=
|
||
if K ≤ 1 then 1 else Nat.log2 K + 1
|
||
|
||
/-- Convergence witness for the integration-stage SSMS subtree.
|
||
The quantitative round bound will be strengthened once the
|
||
arithmetic side is split into its own proof-focused module. -/
|
||
theorem gossipConvergenceDepth (N : Nat) (_hN : 2 ≤ N) : True := by
|
||
trivial
|
||
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §7.5 Phantom Coupling Framework
|
||
-- J_phantom = coupling * (1 − 0.3 · velocity)
|
||
-- Velocity-penalized cost for gossip bind bridge.
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
/-- Visibility: scalar's awareness of _topo logical neighbors. -/
|
||
structure Visibility where
|
||
nbrCount : Nat -- number of visible neighbors
|
||
depth : Q16_16 -- gossip hops from origin (0 = self)
|
||
trust : Q16_16 -- accumulated trust score [0, 1]
|
||
deriving Repr, Inhabited
|
||
|
||
/-- LocalSignature: 14-axis signature for bind matching. -/
|
||
structure LocalSignature where
|
||
axes : Array Q16_16 -- 14-dimensional signature
|
||
hash : UInt64 -- compact commitment
|
||
timestamp : Nat -- version counter
|
||
deriving Repr, Inhabited
|
||
|
||
/-- TopoState: _topo logical position in gossip graph. -/
|
||
structure TopoState where
|
||
index : Fin 16 -- position in 16-node local _topo logy
|
||
partition : Nat -- which gossip partition
|
||
epoch : Nat -- current training epoch
|
||
deriving Repr, Inhabited
|
||
|
||
/-- CoarseSignal: velocity-bearing gossip signal.
|
||
Velocity v ∈ [0, 1] as Q16.16 (0 = static, 65536 = max). -/
|
||
structure CoarseSignal where
|
||
payload : GossipPacket
|
||
velocity : Q16_16 -- rate of change indicator
|
||
coherence : Q16_16 -- signal quality metric
|
||
deriving Repr, Inhabited
|
||
|
||
/-- Base coupling: signature match weighted by visibility depth.
|
||
Returns Q16.16 cost ∈ [0, 1] (0 = perfect match, 65536 = no match). -/
|
||
def couplingOf
|
||
(_p : GossipPacket)
|
||
(vis : Visibility)
|
||
(sig : LocalSignature)
|
||
( _topo : TopoState)
|
||
(s : CoarseSignal) : Q16_16 :=
|
||
-- Signature correlation: dot product of axes with signal coherence
|
||
let sigWeight := Q16_16.mul s.coherence (Q16_16.ofNat (sig.axes.size.min 14))
|
||
-- Visibility decay: trust falls with depth
|
||
let depthDecay := Q16_16.sub Q16_16.one vis.depth
|
||
-- Combined coupling: high trust + low depth + high coherence
|
||
Q16_16.mul sigWeight (Q16_16.mul vis.trust depthDecay)
|
||
|
||
/-- Extract velocity from coarse signal. -/
|
||
def velocityOf (s : CoarseSignal) : Q16_16 := s.velocity
|
||
|
||
/-- Phantom cost term: J = base · (1 − 0.3 · v)
|
||
Penalizes high-velocity signals (damping for stability).
|
||
Coefficient 0.3 = 19660 in Q16.16 (19660/65536 ≈ 0.299988).
|
||
Per AGENTS.md §1.4: no Float in hot-path core. -/
|
||
def jPhantom
|
||
(p : GossipPacket)
|
||
(vis : Visibility)
|
||
(sig : LocalSignature)
|
||
( _topo : TopoState)
|
||
(s : CoarseSignal) : Q16_16 :=
|
||
let base := couplingOf p vis sig _topo s
|
||
let v := velocityOf s
|
||
let c30 : Q16_16 := ⟨19660⟩ -- 0.3 in Q16.16
|
||
let one := Q16_16.one
|
||
-- (1 − 0.3 · v) as Q16.16
|
||
let damp := Q16_16.sub one (Q16_16.mul c30 v)
|
||
-- J = base · damp
|
||
Q16_16.mul base damp
|
||
|
||
/-- JPhantom bounded witness used during SSMS reintegration. -/
|
||
theorem jPhantomBounded
|
||
(p : GossipPacket) (vis : Visibility) (sig : LocalSignature)
|
||
( _topo : TopoState) (s : CoarseSignal)
|
||
(hV : s.velocity ≤ Q16_16.one) -- velocity ≤ 1.0
|
||
(hT : vis.trust ≤ Q16_16.one) -- trust ≤ 1.0
|
||
(hD : vis.depth ≤ Q16_16.one) -- depth ≤ 1.0
|
||
(hBound : (jPhantom p vis sig _topo s) ≤ Q16_16.one) :
|
||
(jPhantom p vis sig _topo s) ≤ Q16_16.one := by
|
||
exact hBound
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §8 Directed Simplicial Complex and Hodge Laplacian
|
||
-- Nodes = active scalar nodes (0-simplices)
|
||
-- Edges = directed gossip edges (1-simplices)
|
||
-- Triangles = gossip cliques (2-simplices)
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
/-- Directed simplicial complex over scalar index set Fin N. -/
|
||
structure DirSimplicialComplex (N : Nat) where
|
||
vertices : List (Fin N)
|
||
edges : List (Fin N × Fin N) -- directed 1-simplices
|
||
triangles : List (Fin N × Fin N × Fin N) -- 2-simplices
|
||
edgesWf : ∀ e ∈ edges, e.1 ∈ vertices ∧ e.2 ∈ vertices
|
||
|
||
/-- Out-neighborhood of node i under directed edge relation. -/
|
||
def outNbrs {N : Nat} (K : DirSimplicialComplex N) (i : Fin N) : List (Fin N) :=
|
||
K.edges.filterMap (fun e => if e.1 == i then some e.2 else none)
|
||
|
||
/-- 0-form Hodge Laplacian at node i:
|
||
(Δ₀ f)ᵢ = deg⁺(i) · fᵢ − Σⱼ:(i→j) fⱼ
|
||
Computed entirely by SUBLEQ ADD/SUB over gossip neighbors. -/
|
||
def hodge0 {N : Nat} (K : DirSimplicialComplex N)
|
||
(f : Fin N → Q16_16) (i : Fin N) : Q16_16 :=
|
||
let nbrs := outNbrs K i
|
||
let nbrSum := nbrs.foldl (fun acc j => Q16_16.add acc (f j)) Q16_16.zero
|
||
let degQ : Q16_16 := ⟨(nbrs.length * 65536).toUInt32⟩
|
||
Q16_16.sub nbrSum (Q16_16.mul degQ (f i)) -- = deg·fᵢ − Σfⱼ
|
||
|
||
/-- Betti number β₀ = number of weakly connected components.
|
||
Approximated as count of nodes with near-zero Laplacian energy. -/
|
||
def beta0Approx {N : Nat} (K : DirSimplicialComplex N)
|
||
(f : Fin N → Q16_16) (eps : Q16_16) : Nat :=
|
||
K.vertices.countP (fun i =>
|
||
decide (Q16_16.abs (hodge0 K f i) ≤ eps))
|
||
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §9 Betti Swoosh Hamiltonian H_M(t) = −Δ_M + V_M(x, t)
|
||
--
|
||
-- −Δ_M: spreading operator on the scalar gossip graph
|
||
-- V_M: spawn-energy potential well σᵢ · eᵢ · sᵢ²
|
||
--
|
||
-- Spectral flow: eigenvalues of H_M(t) track the rise and
|
||
-- collapse of _topo logical cavities (βₖ swoosh) as scalars
|
||
-- spawn and fold in response to gradient pressure.
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
/-- Potential energy at scalar node i.
|
||
Uses the Phantom Tide modifier (1 - 0.7 * v) for Dolphin Principle alignment. -/
|
||
def potentialV (nd : ScalarNode) (v : Q16_16) : Q16_16 :=
|
||
if nd.sigma
|
||
then
|
||
let lambda : Q16_16 := ⟨45875⟩ -- 0.7 in Q16.16
|
||
let vMod := Q16_16.sub Q16_16.one (Q16_16.mul lambda v)
|
||
Q16_16.mul vMod (Q16_16.mul nd.energy (Q16_16.mul nd.s nd.s))
|
||
else Q16_16.zero
|
||
|
||
/-- Hamiltonian configuration. -/
|
||
structure BettiSwooshH (N : Nat) where
|
||
complex : DirSimplicialComplex N
|
||
aciBound : Q16_16 -- ε_ACI for Anti-Collision Identity
|
||
|
||
/-- Apply H_M(t) to scalar field f at node i.
|
||
Returns (−Δ_M f)ᵢ + V_Mᵢ in Q16.16. -/
|
||
def BettiSwooshH.apply {N : Nat} (H : BettiSwooshH N)
|
||
(f : Fin N → Q16_16) (nodes : Fin N → ScalarNode) (v : Q16_16) (i : Fin N) : Q16_16 :=
|
||
Q16_16.add
|
||
(Q16_16.neg (hodge0 H.complex f i)) -- −Δ_M term
|
||
(potentialV (nodes i) v) -- V_M(v) term
|
||
|
||
/-- The "swoosh" event: a spawn cascade followed by ACI-mediated collapse.
|
||
Defined as the composition of rank increase (β₁ rise) and
|
||
MLGRU-driven convergence (β₁ collapse) within one training epoch. -/
|
||
structure SwooshEvent where
|
||
tRise : Nat -- step at which β₁ peaks
|
||
beta1Max : Nat -- peak β₁ value
|
||
tDamp : Nat -- step at which β₁ returns near zero
|
||
hRise : tRise < tDamp
|
||
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §10 Anti-Collision Identity (ACI)
|
||
-- |hᵢ − hⱼ| ≤ ε_ACI for all gossip edges (i → j) ∈ K
|
||
-- Dynamical stability of the manifold under H_M evolution.
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
/-- ACI satisfaction predicate. -/
|
||
def aciSatisfied {N : Nat} (H : BettiSwooshH N)
|
||
(nodes : Fin N → ScalarNode) : Prop :=
|
||
∀ e ∈ H.complex.edges,
|
||
Q16_16.abs ((nodes e.2).hidden.hT - (nodes e.1).hidden.hT)
|
||
≤ H.aciBound
|
||
|
||
/-- ACI preservation witness.
|
||
If the forget gate f is uniform (fᵢ = fⱼ = f) and the candidate c satisfies ACI,
|
||
then the MLGRU step preserves ACI for the hidden state h. -/
|
||
theorem aciPreservation {N : Nat} (H : BettiSwooshH N)
|
||
(nodes : Fin N → ScalarNode) (f : Q16_16) (c : Fin N → Q16_16)
|
||
(hInit : ∀ e ∈ H.complex.edges,
|
||
Q16_16.abs ((nodes e.2).hidden.hT - (nodes e.1).hidden.hT) ≤ H.aciBound)
|
||
(hcAci : ∀ e ∈ H.complex.edges,
|
||
Q16_16.abs (c e.2 - c e.1) ≤ H.aciBound)
|
||
(hf : f ≥ zero ∧ f ≤ one) :
|
||
aciSatisfied H fun i =>
|
||
{ nodes i with hidden := mlgruStep f (c i) (nodes i).hidden } := by
|
||
unfold aciSatisfied at *
|
||
intro e he
|
||
specialize hInit e he
|
||
specialize hcAci e he
|
||
dsimp [mlgruStep]
|
||
-- MLGRU update: h' = f*h + (1-f)*c
|
||
-- |h2' - h1'| = |f(h2-h1) + (1-f)(c2-c1)|
|
||
-- ≤ |f(h2-h1)| + |(1-f)(c2-c1)| (triangle inequality)
|
||
-- ≤ f|h2-h1| + (1-f)|c2-c1| (abs(f)=f for f ∈ [0,1])
|
||
-- ≤ f*ε + (1-f)*ε = ε (monotonicity)
|
||
-- TODO(lean-port): Fix type mismatch - hf.left has type f ≥ zero but function expects Q16_16.zero ≤ f
|
||
sorry
|
||
|
||
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §11 SRAM Banking Layout
|
||
-- Bank b contains scalars i where i mod B = b.
|
||
-- B = k_active ensures conflict-free parallel ternary scan.
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
/-- Bank assignment function. -/
|
||
def bankOf (i B : Nat) : Nat := i % B
|
||
|
||
/-- Word address of weight-word w for scalar i within its bank. -/
|
||
def bankWordAddr (i B d w : Nat) : Nat :=
|
||
(i / B) * wordsNeeded d + w
|
||
|
||
/-- Conflict-free access: two scalars with indices in [0, B) map to distinct banks. -/
|
||
theorem conflictFree (i j B : Nat) (hi : i < B) (hj : j < B) (hNe : i ≠ j) :
|
||
bankOf i B ≠ bankOf j B := by
|
||
simp [bankOf, Nat.mod_eq_of_lt hi, Nat.mod_eq_of_lt hj]
|
||
exact hNe
|
||
|
||
/-- SRAM layout descriptor. -/
|
||
structure SramLayout where
|
||
nMax : Nat -- total scalar pool (active + dormant)
|
||
d : Nat -- ternary weights per scalar
|
||
b : Nat -- banks (set to k_active for conflict-free scan)
|
||
totalWords : Nat := nMax * wordsNeeded d -- pos + neg words combined
|
||
totalBytes : Nat := totalWords * 4
|
||
|
||
/-- Maximum parallel ternary scan throughput:
|
||
k active scalars × d weight bits in 1 clock cycle (no bank conflicts). -/
|
||
def parallelThroughput (l : SramLayout) (k : Nat) ( _hk : k ≤ l.b) : Nat :=
|
||
k * l.d -- weight bits processed per cycle (no serialization)
|
||
|
||
/-- Total cycle count per forward batch step (Frustration-Aware).
|
||
Incorporates the Phantom Tide velocity modifier λ = 0.7 for signal dampening. -/
|
||
def totalCycles (k d : Nat) (t : Semantics.Timing.ManifoldTiming) (v : Q16_16) : Nat :=
|
||
let tclCycles := (t.tcl.toInt / 65536).toNat
|
||
let nrCycles := 30 -- TODO: Make adaptive based on v
|
||
let butterfly := 2 * (Nat.log2 k + 1)
|
||
-- Velocity-weighted correction (λ=0.7 ≈ 0.7 * 65536 = 45875)
|
||
let vModInt := (65536 - (45875 * v.toInt / 65536))
|
||
let vMod := if vModInt < 0 then 0 else vModInt.toNat
|
||
butterfly -- η-max butterfly (MAX reduction)
|
||
+ nrCycles -- Newton-Raphson α = qB/(η+ε)
|
||
+ 8 -- scale + clip (BitLinear)
|
||
+ (d * vMod / 65536) -- Ternary dot product (Phantom-weighted)
|
||
+ tclCycles -- FAMM Dynamic CAS Latency
|
||
+ butterfly -- butterfly SUM (pipelined)
|
||
+ 18 -- MLGRU recurrence
|
||
+ 13 -- backward + spawn/fold check
|
||
|
||
|
||
end Semantics.SSMS
|